Finite Group Databasebeta

← #37511015

Subgroups — #37511015

Order 22912.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26039189C211yesmaximal, characteristic, complemented
2#26038630C212yesmaximal
3#26038616C211yesmaximal, characteristic
4#26038228C212yesmaximal, complemented
5#354×C179C211yesmaximal, characteristic, complemented
6#19153158C2211yescharacteristic
7#19153144—21no—
8#19153111—21no—
9#19153106—22no—
10#19153105—21no—
11#19153097—22no—
12#19153094C2212yes—
13#19153090—22no—
14#19153041—21no—
15#19153030C2211yescharacteristic
16#19153027—22no—
17#124×C179C2211yescharacteristic
18#99×C179C2212yes—
19#15668259—21no—
20#15668258—41no—
21C2×Dic358—21no—
22C2×C22⋊D179—22no—
23D4×D179—41no—
24C22×Dic179D412yes—
25#15668249—41no—
26#15668246—41no—
27#15668243—21no—
28C4⋊Dic179—21no—
29C4⋊Dic179—21no—
30C4⋊Dic179—42no—
31Dic179⋊C4—22no—
32Dic179⋊C4—22no—
33Dic179⋊C4—22no—
34C22⋊Dic179—22no—
35C22⋊Dic179D412yes—
36(C2×D179)⋊C4—42no—
37C4×Dic179—21no—
38C2×Q8×C179D411yescharacteristic
39C2×D4×C179C2311yescharacteristic
40C22⋊C4×C179—22no—
41C22⋊C4×C179—22no—
42C42×C179D411yescharacteristic
43C22×D179—22no—
44Dic358—21no—
45Dic358—81no—
46D716—21no—
47C22⋊D179—42no—
48C22⋊D179—41no—
49C2×Dic179—22no—
50C2×Dic179—22no—
51C2×Dic179—22no—
52C2×Dic179—42no—
53C4×D179—41no—
54C179⋊C8—41no—
55C23×C179C2×D412yes—
56Q8×C179—41no—
57D4×C179—41no—
58C2×C4×C179—21no—
59C2×C4×C179—21no—
60C2×C4×C179—42no—
61C2×C4×C179C2×D411yescharacteristic
62C22×C179—22no—
63C22×C179—42no—
64C22×C179#12011yescharacteristic
65C2×D179—41no—
66C2×D179—82no—
67C4×C179—21no—
68C4×C179—41no—
69C4×C179—41no—
70C4×C179—82no—
71Dic179—42no—
72Dic179—41no—
73Dic179—42no—
74C2×C179—21no—
75C2×C179—42no—
76C2×C179B(4,𝔽2)11yescharacteristic
77D179—81no—
78C179#220111yescharacteristic, complemented, Sylow
79#2201—1791nomaximal, Sylow
80#563—1791no—
81#458—1792no—
82#455—1791no—
83#354D17911yescharacteristic, complemented
84#352—1792no—
85H(5,𝔽2)—1791no—
86#137—3581no—
87C4⋊Q8—3581no—
88#126—3582no—
89#125—3581no—
90#124C2×D17911yescharacteristic
91#123—3582no—
92#123—1792no—
93C22⋊Q8—3582no—
94#104—3581no—
95#100—1791no—
96#99—3582no—
97#99C2×D17912yes—
98#41—3581no—
99#41—7161no—
100C2×Q8—3581no—
101C2×Q8D71611yescharacteristic
102C2×D4—3582no—
103C2×D4—7161no—
104C2×D4C22×D17911yescharacteristic
105C22×C4—1792no—
106Q16—7161no—
107SD16—7161no—
108M16—3581no—
109C4⋊C4—3582no—
110C4⋊C4—3582no—
111C4⋊C4—3581no—
112C4⋊C4—3581no—
113C4⋊C4—3582no—
114C4⋊C4—7162no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—3582no—
118C22⋊C4—1792no—
119C22⋊C4—7162no—
120C42—3581no—
121C42D71611yescharacteristic
122C23—3582no—
123C23D4×D17912yes—
124Q8—41no—
125Q8—3581no—
126Q8—14321no—
127D4—41no—
128D4—3581no—
129D4—7162no—
130D4—7161no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—3582no—
135C2×C4—3582no—
136C2×C4—3582no—
137C2×C4—7161no—
138C2×C4—7162no—
139C2×C4C2×D71611yescharacteristic
140C8—7161no—
141C22—22no—
142C22—42no—
143C22—14322no—
144C22—7161no—
145C22#1915308311yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—82no—
150C4—7162no—
151C4—7161no—
152C4—7162no—
153C2—21no—
154C2—42no—
155C2—14321no—
156C2#2603862711yescharacteristic
subgroupG/H
centerC2#26038627
commutator, cosocleC2×D4×C179C23
FrattiniC2×D4C22×D179
Fitting#354×C179C2
socleC2×C179B(4,𝔽2)
prime radical#19153158C22
hypercenter#354D179
hypocenterC179#2201
LCSG ▷ C2×D4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26039189C2semidirect
4. #26038228C2semidirect
5. #354×C179C2semidirect
78. C179#2201semidirect
83. #354D179semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260391892
#260386302
#260386162
#260382282
#354×C1792
#2201179